Question

Two spheres of equal density and different diameters fall through a liquid with an unknown density and an unknown viscosity. The diameter of the large sphere is twice the diameter of the small sphere. Assume particle Reynolds number is less than one for both spheres. What is the ratio of terminal velocities of large sphere to small sphere?

• A. 1
• B. 2
• C. 4
• D. 0.5

Hint:

Always check the flow regime:
If \( Re_p \lt 1 \) (Stokes' regime), \( u_t \propto d_p^2 \).
If \( Re_p \gt 1000 \) (Newton's regime), \( u_t \propto d_p^{0.5} \).
Knowing these relationships allows quick mental calculation.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This question requires us to determine how the terminal settling velocity of a sphere depends on its diameter in the laminar settling regime.
This is a standard problem in particle mechanics and fluid-particle dynamics.

Step 2: Key Formula or Approach:
The particle Reynolds number is given as less than one (\( Re_p \lt 1 \)), which means the settling occurs in the Stokes' law (laminar) regime.
The formula for Stokes' terminal settling velocity (\( u_t \)) is:
\[ u_t = \frac{g \cdot d_p^2 \cdot (\rho_p - \rho_f)}{18 \cdot \mu} \]
where:
\( g \) is acceleration due to gravity,
\( d_p \) is particle diameter,
\( \rho_p \) is particle density,
\( \rho_f \) is fluid density,
\( \mu \) is fluid viscosity.

Step 3: Detailed Explanation:

• Identify the constant variables:
The density of both spheres is equal: \( \rho_{p1} = \rho_{p2} = \rho_p \).
They are falling through the same fluid, so \( \rho_f \) and \( \mu \) are identical for both.
Gravity \( g \) is constant.

• Establish the proportionality:
Since all other parameters are constant:
\[ u_t \propto d_p^2 \]

• Write the ratio for the two spheres:
Let subscript 1 represent the large sphere and subscript 2 represent the small sphere.
\[ \frac{u_{t1}}{u_{t2}} = \left(\frac{d_{p1}}{d_{p2}}\right)^2 \]

• We are given that the diameter of the large sphere is twice that of the small sphere:
\[ d_{p1} = 2 \cdot d_{p2} \quad \implies \quad \frac{d_{p1}}{d_{p2}} = 2 \]

• Substitute this ratio into the velocity relation:
\[ \frac{u_{t1}}{u_{t2}} = (2)^2 = 4 \]


Step 4: Final Answer:
The ratio of the terminal settling velocity of the large sphere to the small sphere is 4.